LMFDB - Newform orbit 4900.2.a.bb

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(1,4900)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4900.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.1266969904$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 4x + 3$ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$3$
Twist minimal:	no (minimal twist has level 700)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{2} q^{3} + ( - \beta_{2} - \beta_1 + 2) q^{9} + ( - \beta_{2} + 1) q^{11} + ( - \beta_{2} - 3) q^{13} + (\beta_1 - 3) q^{17} + (\beta_1 + 1) q^{19} + (\beta_{2} - \beta_1 - 1) q^{23} + (3 \beta_{2} + \beta_1 - 2) q^{27}+ \cdots + ( - 7 \beta_{2} - 2 \beta_1 + 4) q^{99}+O(q^{100})$ q + b2 * q^3 + (-b2 - b1 + 2) * q^9 + (-b2 + 1) * q^11 + (-b2 - 3) * q^13 + (b1 - 3) * q^17 + (b1 + 1) * q^19 + (b2 - b1 - 1) * q^23 + (3b2 + b1 - 2) q^27 + (2b2 + b1 + 1) q^29 + (-2b2 - b1) q^31 + (2b2 + b1 - 5) q^33 + (-3b2 + 1) q^37 + (-2b2 + b1 - 5) q^39 + (3b2 - b1 - 3) q^41 + (b2 + b1 - 3) * q^43 + (-b2 - b1 - 2) * q^47 + (-6b2 - 3) q^51 + (3b2 + b1 + 6) q^53 + (-2b2 - 3) q^57 + (b2 + 2) * q^59 + (-2b2 + b1 - 6) q^61 + (-2b2 + 6) q^67 + (b2 - b1 + 8) * q^69 + (-2b2 - 7) q^71 - 4 * q^73 + (-3b2 + b1 - 1) q^79 + (-5b2 + 6) q^81 + (-4b2 - b1 - 11) q^83 + (-4b2 - 2b1 + 7) * q^87 + (-b2 - b1 + 4) * q^89 + (5b2 + 2b1 - 7) * q^93 + (-b2 - 2b1 - 6) q^97 + (-7b2 - 2b1 + 4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q - q^{3} + 8 q^{9} + 4 q^{11} - 8 q^{13} - 10 q^{17} + 2 q^{19} - 3 q^{23} - 10 q^{27} + 3 q^{31} - 18 q^{33} + 6 q^{37} - 14 q^{39} - 11 q^{41} - 11 q^{43} - 4 q^{47} - 3 q^{51} + 14 q^{53} - 7 q^{57}+ \cdots + 21 q^{99}+O(q^{100})$ 3 * q - q^3 + 8 * q^9 + 4 * q^11 - 8 * q^13 - 10 * q^17 + 2 * q^19 - 3 * q^23 - 10 * q^27 + 3 * q^31 - 18 * q^33 + 6 * q^37 - 14 * q^39 - 11 * q^41 - 11 * q^43 - 4 * q^47 - 3 * q^51 + 14 * q^53 - 7 * q^57 + 5 * q^59 - 17 * q^61 + 20 * q^67 + 24 * q^69 - 19 * q^71 - 12 * q^73 - q^79 + 23 * q^81 - 28 * q^83 + 27 * q^87 + 14 * q^89 - 28 * q^93 - 15 * q^97 + 21 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{3} - x^{2} - 4x + 3$ x^3 - x^2 - 4*x + 3 :

$\beta_{1}$	$=$	$\nu^{2} + 2\nu - 4$ v^2 + 2*v - 4
$\beta_{2}$	$=$	$\nu^{2} - \nu - 3$ v^2 - v - 3

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( -\beta_{2} + \beta _1 + 1 ) / 3$ (-b2 + b1 + 1) / 3
$\nu^{2}$	$=$	$( 2\beta_{2} + \beta _1 + 10 ) / 3$ (2*b2 + b1 + 10) / 3

Display $\beta_i$ in terms of $\nu^j$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

0.713538
2.19869
−1.91223

−3.20440

7.26819

1.2

−0.364448

−2.86718

1.3

2.56885

3.59899

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$+1$
$7$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4900.2.a.bb		3
5.b	even	2	1		4900.2.a.bd		3
5.c	odd	4	2		4900.2.e.s		6
7.b	odd	2	1		4900.2.a.bc		3
7.d	odd	6	2		700.2.i.d	✓	6
35.c	odd	2	1		4900.2.a.ba		3
35.f	even	4	2		4900.2.e.t		6
35.i	odd	6	2		700.2.i.e	yes	6
35.k	even	12	4		700.2.r.d		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
700.2.i.d	✓	6	7.d	odd	6	2
700.2.i.e	yes	6	35.i	odd	6	2
700.2.r.d		12	35.k	even	12	4
4900.2.a.ba		3	35.c	odd	2	1
4900.2.a.bb		3	1.a	even	1	1	trivial
4900.2.a.bc		3	7.b	odd	2	1
4900.2.a.bd		3	5.b	even	2	1
4900.2.e.s		6	5.c	odd	4	2
4900.2.e.t		6	35.f	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4900))$ :

T_{3}^{3} + T_{3}^{2} - 8T_{3} - 3

T_{11}^{3} - 4T_{11}^{2} - 3T_{11} + 9

T_{13}^{3} + 8T_{13}^{2} + 13T_{13} - 3

T_{19}^{3} - 2T_{19}^{2} - 23T_{19} - 21

T_{23}^{3} + 3T_{23}^{2} - 36T_{23} - 81

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3} + T^{2} - 8T - 3$ T^3 + T^2 - 8*T - 3
$5$	$T^{3}$ T^3
$7$	$T^{3}$ T^3
$11$	$T^{3} - 4 T^{2} + \cdots + 9$ T^3 - 4T^2 - 3T + 9
$13$	$T^{3} + 8 T^{2} + \cdots - 3$ T^3 + 8T^2 + 13T - 3
$17$	$T^{3} + 10 T^{2} + \cdots - 81$ T^3 + 10T^2 + 9T - 81
$19$	$T^{3} - 2 T^{2} + \cdots - 21$ T^3 - 2T^2 - 23T - 21
$23$	$T^{3} + 3 T^{2} + \cdots - 81$ T^3 + 3T^2 - 36T - 81
$29$	$T^{3} - 45T + 81$ T^3 - 45*T + 81
$31$	$T^{3} - 3 T^{2} + \cdots - 37$ T^3 - 3T^2 - 42T - 37
$37$	$T^{3} - 6 T^{2} + \cdots + 149$ T^3 - 6T^2 - 63T + 149
$41$	$T^{3} + 11 T^{2} + \cdots - 873$ T^3 + 11T^2 - 78T - 873
$43$	$T^{3} + 11 T^{2} + \cdots - 71$ T^3 + 11T^2 + 14T - 71
$47$	$T^{3} + 4 T^{2} + \cdots - 9$ T^3 + 4T^2 - 21T - 9
$53$	$T^{3} - 14 T^{2} + \cdots + 549$ T^3 - 14T^2 - 15T + 549
$59$	$T^{3} - 5T^{2} + 9$ T^3 - 5*T^2 + 9
$61$	$T^{3} + 17 T^{2} + \cdots + 1$ T^3 + 17T^2 + 26T + 1
$67$	$T^{3} - 20 T^{2} + \cdots - 72$ T^3 - 20T^2 + 100T - 72
$71$	$T^{3} + 19 T^{2} + \cdots + 45$ T^3 + 19T^2 + 87T + 45
$73$	$(T + 4)^{3}$ (T + 4)^3
$79$	$T^{3} + T^{2} + \cdots + 449$ T^3 + T^2 - 118*T + 449
$83$	$T^{3} + 28 T^{2} + \cdots - 981$ T^3 + 28T^2 + 129T - 981
$89$	$T^{3} - 14 T^{2} + \cdots + 45$ T^3 - 14T^2 + 39T + 45
$97$	$T^{3} + 15 T^{2} + \cdots - 5$ T^3 + 15T^2 - 18T - 5
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